Document o8emrJkoqpr3Zpn6jXL20qYo

76 CHAPTER 4 1954 Guide solid and partly broken) of Fig. 6 is found. The maximum value of may be computed by differentiating w with respect to p and equating the result to zero. This operation produces the formula: R / 2 \i (40) i< 4 i. "For air, with k = 1.40, -- = 0.53. Pi Actually, the broken part of the curve is not attained for the flow in the .nozzle. If the ratio of pi to pi is decreased from unity, the mass rate of discharge, as well as the volume, increases from zero to a maximum, as shown by the solid section of the curve in Fig. 6; thereafter, as pi/pi is decreased further, the discharge is constant, as indicated by the horizontal line. The value of pj at the maximum point is called the critical pressure, or Po, and it is seen that p, is approximately 53 percent of pi when air is flowing. k g X I i Fig. 6. Relation or Flow or Gas to Pkbssube Drop in a Convebging Tube I To find the velocity at the critical pressure, it is assumed that the upstream velocity Vi is so small as to be negligible. Using the subscript c to indicate conditions at the critical point, from Equation 20 Fluid Flow 77 In developing the working equations for orifices and nozzles, it is custom ary. to start with the incompressible form of the flow Equation 38. In this case both ikf, and Mi are small quantities, .pi = pi, and (pl -- pj)/p> = Ap/pi is small. Retaining only first order terms, it follows from Equation 35 that so that Mi At Mt ~ h (44) ' k-1 1+---W 1 - MS/M,* Vl - (At/A,) , vT (45) where 0 _ A/D,. The quantity l/Vl - 0* is the velocity of approach -J 0.80, I 070 O 50.02 Oj ^ " 50lI0D, w 0.68 oW 030 ilrr UQ$003 0,--J 0.03 D, 4-IOJ JO4 2 5 10* 2 pV|Di 020 0.10 3 10* 0.03 (O) Fig. 7. Dimensions and Flow Coefficient for Standard Sharp Edged Orifice (Coefficient shown as a function of Reynolds Number and Ratio, A/A.) Note: From Reference 4. Used by permission. factor as generally used, with 0 being the ratio of the throat or orifice di ameter to the pipe diameter. Since Ap/p2 is small Ior Substituting the critical pressure ratio from Equation 40 it follows that M,= 1 (43) or that the velocity at the throat is equal to the local sonic velocity. and the mass flow is w = VT^ The volume flow is then 0 = At V^Zp/p =. At W7) ]. (48)